Nonparametric procedures for a better applicability of the L-moment homogeneity test

Abstract

To estimate quantiles (representing the hydrologic risk) at an ungauged site, regional frequency analysis (RFA) is commonly used. To provide accurate estimates, in RFA regions must be homogeneous. Thus, the first step of such an analysis is to check this homogeneity, usually through the Hosking-Wallis (HW) homogeneity test. Though this test is useful and powerful, it presents some drawbacks which include the necessity of estimating a subjectively chosen parametric distribution and a poorly justified rejection threshold. In the present work, these drawbacks are addressed through the use of a nonparametric framework for the HW test. Thus, three resampling methods are considered to obtain the rejection threshold without requiring any prior distribution. In addition, the computation of a statistically justified p-value is proposed instead of the original rejection threshold. The three nonparametric methods and the original HW test are compared through a simulation study. Results show that permutation methods and the bootstrap are more powerful than the original HW test. The nonparametric tests are also easier to implement and needs less time to perform.

Full text

Nonparametric procedures for a
better applicability of the L-moment
homogeneity test
25 mai 2016
CWRA 69th national conference
Pierre Masselot, Fateh Chebana, Taha B.M.J. Ouarda

Contents
1. Introduction
2. The Hosking-Wallis homogeneity test
3. Proposed improvements
4. Simulation study
5. Conclusion
Introduction HW test Improvements Sim. Study Conclusion 2

Regional frequency analysis
Frequency analysis (FA):
→Fit a parametric probability distribution to data
→Estimate frequency of extreme events from this distribution
Regional frequency analysis (RFA)
→Perform frequency analysis on a set of sites (a “region”)
→Assess the characteristics of ungauged (or partially gauged) sites
→Necessitates the homogeneity of the region
Required preliminary step of RFA
Test for the homogeneity of a region
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Introduction HW test Improvements Sim. Study Conclusion

Homogeneity testing
Preliminary phase of RFA
Goal:
→decide whether or not a set of sites (a region) can be considered
homogeneous
Important step:
→The accuracy of estimated quantiles depends on it
Most used test:
The Hosking-Wallis (HW) homogeneity test
(Hosking and Wallis 1993)
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Introduction HW test Improvements Sim. Study Conclusion

The Hosking-Wallis Homogeneity test
1. Compute the test statistic
→: L-scale
→ is the weighted variance of at-site scale measures
→The larger is, the less homogeneous the regions is
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Introduction HW test Improvements Sim. Study Conclusion

The Hosking-Wallis Homogeneity test
1. Compute the test statistic
2. Simulate a set of homogeneous region
→All sites are simulated from a 4-parameters Kappa distribution
→Very general distribution
→Contains Normal, Gumbel, GEV distributions as special cases
→The parameters are estimated on observed data
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Introduction HW test Improvements Sim. Study Conclusion

The Hosking-Wallis Homogeneity test
1. Compute the test statistic
2. Simulate a set of homogeneous region
3. Compute the statistic for all the regions
→Estimation of the distribution for homogeneous regions
→: mean of the distribution
→: standard deviation of the distribution
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Introduction HW test Improvements Sim. Study Conclusion

The Hosking-Wallis Homogeneity test
1. Compute the test statistic
2. Simulate a set of homogeneous region
3. Compute the statistic for all the regions
4. Compute the heterogeneity measure
→: homogeneous
→: possibly homogeneous
→: heterogeneous
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Introduction HW test Improvements Sim. Study Conclusion

Drawbacks
1. Compute the test statistic
2. Simulate a set of homogeneous region
3. Compute the statistic for all the regions
4. Compute the heterogeneity measure
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Introduction HW test Improvements Sim. Study Conclusion

Drawbacks
1. Compute the test statistic
2. Simulate a set of homogeneous region
The use of a parametric distribution creates uncertainty
→.Relevant only if data follow the distribution
→.Necessitates the estimation of 4 parameters
→.Issue more important in the multivariate case
Proposition:
Simulate the regions through nonparametric procedures
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Introduction HW test Improvements Sim. Study Conclusion

Drawbacks
1. Compute the test statistic
2. Simulate a set of homogeneous region
3. Compute the statistic for all the regions
4. Compute the heterogeneity measure
→.Rejection threshold not well justified
Proposition:
Compute a p-value from the distribution
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Introduction HW test Improvements Sim. Study Conclusion

Step 2: nonparametric procedure
Simulate homogeneous regions from the empirical distribution
→i.e. from the pooling of all sites data
M: Permutations methods (Fisher 1935; Pitman 1937)
→Randomly reassigning the observation between sites
→Same as sampling sites without replacement from
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Introduction HW test Improvements Sim. Study Conclusion

Step 2: nonparametric procedure
Simulate homogeneous regions from the empirical distribution
→i.e. from the pooling of all sites data
M: Permutations methods (Fisher 1935; Pitman 1937)
B: Bootstrap (Efron 1979)
→Sampling with replacement from
→Allows testing more general hypotheses than permutations
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Introduction HW test Improvements Sim. Study Conclusion

Step 2: nonparametric procedure
Simulate homogeneous regions from the empirical distribution
→i.e. from the pooling of all sites data
M: Permutations methods (Fisher 1935; Pitman 1937)
B: Bootstrap (Efron 1979)
Y: Pólya resampling (Lo 1988)
→Bootstrap without the assumption that the empirical distribution
represents the true distribution of
→Each time an observation is drawn, a new one is added to
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Introduction HW test Improvements Sim. Study Conclusion

Step 2: nonparametric procedure (illustration)
15
Empty
MBY
Step 1
Step 2
Introduction HW test Improvements Sim. Study Conclusion

Step 4: rejection threshold
After step 3: set of simulated at disposal
→Represents the distribution of under homogeneity
A uses the whole distribution
Compare the to a chosen significance level
→If : reject the hypothesis of homogeneity
→If : do not reject the hypothesis of homogeneity
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Introduction HW test Improvements Sim. Study Conclusion

Simulation study
Goal: compare the HW test and the nonparametric procedures
1. Creation of artificial regions with identical known characteristics ;
→Bivariate regions (peak and volume, Chebana and Ouarda 2007) ;
→Homogeneous or heterogeneous ;
→With different number of sites.
2. Application of the test on each of the regions ;
→Results in decisions ;
3. Evaluation of the performances through the rejection rate.
→Type I error: proportion of homogeneous regions rejection ;
→Power: proportion of heterogeneous regions rejection ;
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Introduction HW test Improvements Sim. Study Conclusion

Results: type I error
Regions simulated: homogeneous
→Rejection rate must be close to the significance level
→ HW and Y tests: underestimated type I error
→ M and B tests: type I error closer to
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Introduction HW test Improvements Sim. Study Conclusion

Results: power
Regions simulated: heterogeneous
→We want the power to be as high as possible
→ Y test has a low power
→ M and B tests outperform the HW test
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Introduction HW test Improvements Sim. Study Conclusion

Conclusion
In order to improve the HW test we propose:
→To use nonparametric procedures to simulate the distribution
→To compute a to decide to reject the homogeneity or not
This leads to:
→A wider applicability of the test
→A simplification of the test procedure
→An increase of the power of the test (M and B procedures only)
Among the three procedure M and B are more powerful than Y
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Introduction HW test Improvements Sim. Study Conclusion

Thank you for listening
●Masselot, P., Chebana, F., & Ouarda, T. B. M. J. (2016). Fast and
direct nonparametric procedures in the L-moment homogeneity
test. Stochastic Environmental Research and Risk Assessment,
1-14. doi: 10.1007/s00477-016-1248-0
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Introduction HW test Improvements Sim. Study Conclusion

L-moments (Hosking 1990)
Alternative to classical moments
→More robust than classical moments
→Based on order statistics
Formally, for continuous distributions:
→: quantile function
→: shifted Legendre polynomials
L-moments are linear combinations of a distribution’s quantiles
Introduction HW test Application Conclusion 22

L-moments (Hosking 1990)
Alternative to classical moments
→More robust than classical moments
→Based on order statistics
L-moments are linear combinations of quantiles
Examples:
→L-scale
→Mean difference between two observations
→L-skewness
→Tells if the central observation is closer than one the two extrema
Introduction HW test Application Conclusion 23

Hosking-Wallis test statistic
Based on the L-CV
→Dimensionless scale measure
→Most important distribution characteristic in RFA
HW test statistic:
→: weighted variance of at-sites
→HW test is a kind of ANOVA
The larger is, the more different at-site distributions are
Introduction HW test Application Conclusion 24